Is the electric going on and off while appliances are in use?
Regardless of the age of the home, if there wasn't some thought put into the wiring of the home, you can often have more major appliances running on one line than you should.
I once rented a home to someone who liked to run the dishwasher, washer, dryer and microwave all at the same time. These appliances happen to all be on the same line in the house and one day *pop* went the fuse. The electrician explained that microwaves pull a lot of juice, and hers happened to be faulty, so that was probably what it did it. As soon as we tested her microwave, it was shooting sparks. The renter knew it too, but blamed each landlord instead of the common denominator - her microwave

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Most older buildings have had their electric rewired at some point. You can tell by looking at the fuse box (check the amp rating and you also might find a date listed).
Fridges can also run your bill up. Maybe the coils are dusty? Try vacuuming the bottom and back of the fridge. Once you clean it, it will run more efficiently. You can check the setting on the fridge/freezer too (maybe it's set too cold).
JEA will come out and look at the meter for you and check your apartment to see where else you might find savings. I'd also want to see the JEA bill every month, your landlord should be giving you a copy.
The on and off of the electric is a safety concern though, I'd want the landlord to call an electrician out and find the source of the problem.
As for your rights, there should be something spelled out in your lease for how the utilities will be billed. Is it possible the landlord is just dividing the entire buildings usage by 4? If so, was that agreed upon in the lease?